Chapter Collisions of Charged Particles 6

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Chapter 6
Collisions of Charged Particles
The interactions of a moving charged particle with any surrounding matter are governed
by the properties of collisions. We will usually call the incident particle the "projectile"
and the components of the matter with which it is interacting the "target-particles" or just
the "targets". The simplest situation one might imagine is that the matter consisted of free
charged particles, electrons and nuclei. This is exactly the situation that applies if the matter
with which the particle is interacting is a plasma. It might be thought that in this case,
the mutual interaction of the target-particles themselves could be ignored, and the collisions
treated as if they were all simple two-body collisions. This is not quite true because of the
long-range nature of the electromagnetic force, as we shall see, but it is possible, nevertheless,
to treat the collisions as two-body, but correct for the influence of the other target particles
in this process.
In interactions with the atoms of solids, liquids or (neutral) gases, the fact that the target
electrons are bound to the nuclei of their atom is obviously, in the end, important to the
interaction processes. The atoms themselves can usually be treated ignoring the interactions between them, a t least for projectiles with substantial kinetic energy. The simplest
approximate analysis goes further, and starts from the highly simplified view that the electrons can be treated initially zgnorzng the force binding them to atoms. The corrections
to this approach are naturally substantial, and the treatment cannot always yield accurate
results. Nevertheless it represents a kind of baseline that more accurate calculations and
measurements can be compared with.
The nuclei of the target are important in collisions with plasmas. However, in interactions
with neutral atoms, direct electromagnetic interaction with the nucleus requires the projectile
to penetrate the shielding of the orbiting electrons in the atom. Only particles with very
high momentum can do that. Therefore the electrons of the target are usually the most
important to consider, and tend to dominate the energy loss.
The topic of atomic collisions is an immense and complex one, in which quantum mechanics naturally plays a crucial role. It would take us far beyond the present intention to
attempt a proper introduction to this topic. Two simplifying factors enable us, nevertheless,
to develop this aspect of electromagnetic interactions in enough detail for many practical
purposes. The first factor is that the details of atomic structure become far less influential
in collisions at energies much higher than the binding energies of atoms (which is about ten
electron volts or so). The second is that even when quantum effects are important in the
collisions, approximate formulas with wide applicability, but ignoring the details of particular atomic species, can be obtained by semi-classical arguments. The quantum corrections
are then applied in a way that seems somewhat ad hoe, but often represents the way the
earliest calculations were done, and gives simple analytic formulas.
6.1
6.1.1
Elastic Collisions
Reference Frames and Collision Angles
Consider an idealized non-relativistic collision of two interacting particles, subscripts 1 and
2, with positions r l , ~
a nd velocities v l , ~which
,
are not acted on by any forces other than
their mutual interactions and which experience no changes in internal energy, so the collision
is elastic. Their total (combined) momentum, mlvl+m2v2, is constant, so that their centerof-mass,
R E m l r l + m2r2
(6.1)
m l + m2
moves at a constant velocity, the center-of-mass velocity:
It is helpful also to introduce the notation
for what is called the "reduced mass". In terms of this quantity and the relative position
vector r E r l r 2 , the positions of the particles can be written:
-
and their velocities:
m,
m,
v 1 = V + v
v 2 = V v ,
(6.5)
ml
m2
where v E r is the relative velocity.
Some of our calculations need to be done in the center-of-mass frame of reference, in which
R is stationary. Others need t o be done in the lab frame or other frames of reference, for
example in which one or other of the particles is initially stationary. The angles of vectors
in these frames are important. The directions of all position vectors and of all velocity
differences are the same in all inertial frames. However the directions of velocities are not
the same in different frames.
For example, consider a collision illustrated in Fig 6.1. Collisions can be considered in
a single plane-of-scattering which is perpendicular to the angular momentum of the system,
itself a constant. The angle of scattering, which we denote x is just the angle between the
initial direction of the relative velocity v and its final direction, v'. This angle is different
in different reference frames. Call the angle in the center-of-mass frame x,. By conservation
of energy, the final relative velocity v' has absolute magnitude equal to that of the initial
relative velocity, vo. So the final velocity can be written in component form, in the center of
mass frame, as
v 1 = vo(cos xC,sin x,) ,
(6.6)
where we have chosen the initial relative velocity direction for the x-axis
Center-of-Mass Frame
/
Laboratory Frame, Stationary Target -
Particle 1
1
.. !
X, ...'.,:
. . . . . . . . . . . . . . . . . . . . . . . . . . . vo
~
Particle 2
Figure 6.1: Collisions in center-of-mass and laboratory frames
Substituting into eq(6.5) we find the final velocities in the lab frame to be given by
vlI
v'2 -
mr
ml
m,
m2
=v
- --
I
+V
=
m, o sin X ,
+ V,vml
) ,
m,
vo cos xC + V ,v
o sin xC
m2
)
cos X ,
.
The angle in the lab frame of the final velocity of particle 1 to its initial velocity (which is
in the x-direction) , x1 say, is then just given by the ratio of the components of the final
cot X1
-v0
COS XC
= mi
+v
sin xC
?VO
nu
-
cot xc
+ voV mm rl csc X,
(6.9)
-
For the specific case when particle 2 is a statzonary target, with initial lab-frame velocity
zero, the center-of-mass velocity is V = mlvo/(ml m2) = (mr/m2)v0 and so
+
cot XI
X,
= cot
+ ml
csc X,
m2
(6.10)
-
We often want t o know how much energy or momentum is transferred from an incident
projectile, (particle 1) to an initially stationary target (particle 2). Clearly from eq(6.8)
we can obtain these quantities in terms of the scattering angle x,. So, the change in the
x-momentum of particle 1 is simply
+
cos xC V
i
-
mvo
= m,vo
I
cos X,
+
-
(6.11)
and the final recoil energy of particle 2 (which is the energy lost by particle 1) is
Q
1
m
2 z
[
m,
m2
1
2
(
cOs*,
+v +
m,
2
o
0 sin x,
v
i2 *,I +
;:
[ ( cos
1)'
1 4 2
-2vm2
02(1
i2]
+ sin2xCj
-
cos x,)
=
=
=
1
4
v2 m20
2
2 x c
4sin (-)2
Notice that the maximum possible energy transfer, which occurs when X,
=
. (6.12)
18O0,is
All of these relations are completely independent of the nature of the interaction between
the particles, since we have invoked only conservation of momentum and energy.
Impact Parameter and Cross-section
By definition, the cross-section, 0, for any specified collision process, when a particle is
passing though a density n2 of targets, is that quantity which makes the number of such
collisions per unit path length equal to n20.1 Sometimes a continuum of types of collision is
under consideration. For example we can consider collisions giving rise to different scattering
angles (x) t o be distinct. In that case, we speak in terms of differential cross-sections,
and define the differential cross-section $ (for example) as being that quantity such that
the number of collisions within an angle element d x per unit path length is
dff
r(.zzdx
~
~
/
L
'An alternative definition can be invoked, equivalent to this first definition but in the frame of reference
in which the single particle (1) is stationary and the particles of density n s are moving.
Sometimes other authors use different notation for the differential cross-section, for example
~ ( x )However,
.
our notation, with which we are familiar from calculus, is highly suggestive
and the cross-sections obey natural rules for differentials implied by the notation.
For classical collisions, the impact parameter, b, shown in Fig 6.1, is a convenient
parameter by which to characterize the collision. It is the distance of closest approach
that would occur for the colliding particles if they just followed their initial straight-line
trajectories. Alternatively, the impact parameter can be considered to be a measure of the
angular momentum of the system in the center-of-mass frame, which is m,vob.
.
d
l
-
-
. .'
Figure 6.2: Differential volume for counting the number of collisions in length d! with impact
parameter b.
The differential cross-section with respect to the impact parameter is defined purely by
geometry. As illustrated in Fig 6.2, one can think of the projectile (particle 1) as dragging
along with itself an anulus of radius b and thickness db as it moves along a distance d! of
its path length. This anulus drags out a volume d!27rbdb, and the number of targets that
are in this volume, and hence have been encountered in the impact parameter element db
at b in this path-length is n2d!27rbdb. Consequently, from our definition, the differential
cross-section for scattering at impact parameter b is
Notice that the integral of this quantity over all impact parameters (i.e. 0 < b < oo) will
certainly diverge, because it considers the projectile to be colliding with all the target particles it passes, no matter how far away they are. Therefore the total number of "collisions" of
all possible types, per unit length in an infinite target medium is infinite. This mathematical singularity in the "total cross-section" points out the need to define more closely what
constitutes a collision, and alerts us to the fact that for collisions governed by interactions
of infinite range, such as the forces between charged particles, we shall have to define our
collisions in such a way as to account for some effective termination of the impact-parameter
integration.' This termination, which is often expressed approximately as a cut off of the
'It also shows the fundamental incoherence of the notion of the total number of collisions per unit length
and concepts that depend on it such as the average change in some parameter per collision, which some
authors unfortunately employ
impact parameter integration at a maximum b,
will be governed by consideration of the
particle parameter whose change due to collisions we are trying t o calculate. For example,
the momentum or energy change in the collision may become negligible for b > b.,
There is usually a one-to-one relationship between the impact parameter and the angle of
scattering and hence with the energy transfer, Q, given by eq(6.12). Consequently the differential cross-section with respect to energy transfer, scattering angle and impact parameter
are all related thus:
do
d o dxC d o db dx,
-
- -
d Q = d x . d Q = i l b ~ ~ ~
(6.15)
If we are concerned with a quantity such as the energy of the projectile, which is changing
because of collisions, and the change in each collision is an amount Q(b) that depends on
the impact parameter, then the total rate of change per unit length due to all possible types
of collisions is obtained as
n 2 Qdo = Q Q 2nbdb
(6.16)
/
6.1.2
/
Classical Coulomb Collisions
The exact relationship between the impact parameter, b, and the scattering angle is determined by the force field existing between the colliding particles. For electromagnetic
interactions of charged particles, the fundamental force is the Coulomb interaction between
the forces, an inverse square law. As Isaac Newton showed, the orbit of a particle moving
under an inverse square law force is a conic section; that is, an ellipse for closed orbits or a
hyperbola for the open orbits relevant to collisions.
Elementary analysis shows that the resulting scattering angle X, for a collision with
impact parameter b is given by
b
(6.17)
cot
=
,
(?)
-
-
b90
where, for particles of charge ql and q2 and initial collision velocity vo the quantity bgO is
given by
bgo = q1q2 1 2 .
(6.18)
4nto m,vo
Clearly from eq(6.17), bgO is the impact parameter at which the scattering angle in the center
of mass frame is 90". Trignometric identities allow us t o deduce immediately from eq(6.17)
that
1
db
b90
~ i n ' ( ~ ~ /=2 )
and
= - c ~ c ~ ( ~ , / 2 ).
(6.19)
1 (b/bgo)'
dxC
2
So that the energy transfer in a collision (see eq(6.12)) is
-
+
-
and the rate of transfer of energy per unit length for a particle of energy K
with stationary targets is
=~
m l colliding
v ~
where the upper limit of the b-integration, b,
which prevents the integral diverging, will
be discussed in a moment. One way to think of this equation is t o regard the quantity
as an effective collision cross-section for total energy loss. When
7rbg0& l n [ l + (b,,/bg0)2]
multiplied by the density n2 of targets it gives the inverse scale-length for energy loss,
d i n Kid!.
Figure 6.3: Scattering angle and impact parameter shown schematically for different
Coulomb collisions.
The integral over impact parameters diverges if we extend it to infinite b. This is because
the inverse square law has essentially infinite range. As a result, the dominant contribution
to the energy loss cross-section comes from distant collisions, in which b >> bgO, and hence
the scattering angle is small. Several different physical effects can enter a t large impact
parameters to change the effective force-law and prevent the divergence. We will treat these
effects separately in later sections, but in almost all cases, the exact value of the upper limit is
is large and appears
not a very strong quantitative effect on the cross-section because b,,/bgo
inside a logarithmic term that may be written approximately ln(bm/bgO), which therefore
varies very slowly with bm. Many treatments adopt a small-angle approximation for the
differential cross-section earlier in the derivation, leading to an expression Q K l / b 2 and an
integral that diverges both at small b and at large b. Such treatments then need to invoke a
b,
cut-off of the integration, justifying it on the basis of a breakdown of the approximation,
and naturally adopt bgO as that cut-off in this classical case. The resulting expression is
then essentially identical t o ours, which was obtained more rigorously. There are, in some
circumstances, important physical effects that require us to cut-off the integration a t small
b even before bgO is reached. In those cases we simply replace the term ln[l (bm/bg0)2]
with 2 ln(bm/b,,).
+
6.2
Inelastic Collisions
The effects that give rise to the cut-off of the Coulomb logarithm are primarily associated
with the presence of other particles and forces in the system. If the target particles experience
the force-field of another nearby particle, such as will be the case if the targets are electrons
bound to the nuclei t o form the atoms of a target material, then the dynamics of their binding
gives rise to a cut-off. One way to think of this effect is to regard the electrons as behaving
as if they were free only in collisions in which the energy transfer from the projectile is larger
than their binding energy in the atom. Distant, small angle, collisions transfer less energy.
A cut-off b,,
should be applied at that impact parameter where the energy transfer is equal
to approximately the binding energy.
Alternatively, and more physically, one can regard these collisions as being with a composite target system, the atom, in which there is a transfer of energy inelastically t o the
system, the energy being partially taken up in the ionization or excitation energy of the
atom. Clearly, a fully rigorous calculation of such collisions requires the quantum structure
of the atom to be considered, and so is intrinsically quantum-mechanical. Nevertheless, semiclassical calculations, taking quantum effects into account in a somewhat ad hoe manner, give
substantial insight into the governing principles and, in fact, are able to give quantitatively
correct forms for the cross-sections and energy loss.
xcited Electron
Figure 6.4: Collisions with an atomic system can excite or eject electrons from the atom
6.2.1 Energy transfer t o an oscillating particle
An approach to the problem of collisions with bound particles that can be treated classically,
and becomes the basis for a quantum description, is to approximate the system as a charge
bound in a simple harmonic potential well. Because we are mostly interested in large impact
parameters, we regard the electric field of the projectile as uniform at the atom and then
ask the question, in the encounter of the projectile with this oscillating electron, how much
energy does the oscillator gain as a result of the fluctuating electric field of the passing
projectile.
So consider a simple oscillating particle in a uniform electric field, E ( t ) . Its position x is
governed by the equation
4
2 + w2, = E ( t )
(6.22)
m
We solve this equation in the time range (tl, t2), with some assumed initial condition a t tl
so as to determine the energy gained by the particle at time t2. This solution is readily
obtained using what is called the "one-sided Green's function" as follows. The solutions to
the homogeneous problem (the equation with zero right hand side) are sinwt and coswt.
The Green's function is constructed as
H ( t , T)
=
(sin wt cos WT
-
cos wt s i n w ~ ) / w
and the general solution is then
x(t)
= Asinwt
+ B coswt + i H ( ~ , mT4 ) E ( T ) ~,T
(6.24)
t1
where A and B are constants determined by the initial conditions. Actually, we don't need
to solve for A and B because when we calculate the energy of the oscillator, averaged over an
oscillator period, A and B make exactly the same contribution at the end of the integration
as they did at the beginning and any cross-terms between them and H average to zero. [The
point about the cross terms is not really obvious, but in the interests of time we won't prove
it.] Therefore, the gain in energy is determined just by the integral term and we can simply
set A = B = 0. Then at time t2 the solution may be written
wm
x ( t 2 ) = sinwt2
4
7
cos W T E ( T ) ~cos
T wt2
s i n w ~ E ( ~ ) .d ~
-
t1
(6.25)
t1
When this expression is differentiated, the terms arising from the differentials of the limits
cancel and we get
wm .
x ( t 2 ) = w cos wt2
4
7
+
cos W T E ( T ) ~wTsinwt2
tl
7
tl
sin W T E ( T ) ~
. T
(6.26)
So the total (kinetic plus potential) energy in the oscillator can then rapidly be evaluated as
with the Fourier transform of the electric field written
E(w)
= { ~ X ~ ( ~ W T ) E ( T ) ~. T
(6.28)
We did this integration over a finite time, which avoids some mathematical difficulties,
but we can now readily let tl i o o and t2 i oo and obtain the full domain Fourier
integral. We have obtained the important general result that the energy transferred to a
harmonic oscillator is proportional to the Fourier transform of the electric field evaluated at
the resonant frequency of the oscillator, eq(6.27).
6.2.2
Straight-Line Collision
We are interested mostly in small-angle collisions, because, as we previously noted, they
We approximate the orbit of the
dominate the behavior, especially a t the cut-off, b.,
projectile in this case as a straight-line. Then, as illustrated in Fig 6.5, the electric field at
Straight-Line Collision
Figure 6.5: The approximation of a straight orbit gives a simple expression for the electric
field as a function of time.
the atom is just that due to a charge moving past at an impact parameter b and a constant
speed. For a non-relativistic speed v the components of the electric field as a function of
time are then
b
vt
E,(t) = -41
and E, (t) = -41
47rto (b2 + v2t2)3/2
47rto (b2 + v2t2)3/2
----
----
the relativistic forms are qualitatively similar, and were calculated previously in section 4.2,
see eq(4.40)
E,(t)
=
-41
4 r t o (b2
---
yvt
yb
and E,(t) = -41
+ ~ ~ v ~ t ~ ) ~ / 4 r~t o (b2 + ~ ~ v ~ ' t ~ ) ~ / ~
---
where y is the relativistic factor (1 v2/~2)-1/2. The field components are plotted as a
function of time in Fig 6.6. Clearly, by inspection of Fig 6.6, and eq(6.28) there will be a
qualitative change in the behaviour of the Fourier transform of E ( t ) and hence the energy
transfer for wblyv >> 1 compared with wblyv << 1. The characteristic time duration of the
blyv. If this is much shorter than the characteristic oscillator time, l/w, we
collision is
can take w 0 and obtain by elementary integration
-
---
E, (w) = 2 4 1 .
(6.31)
471tobU
Because E,(t) is antisymmetric, E,(w) = 0 in this small impact parameter limit. In the
opposite limit, that is for collisions in which b is so large that wblyv >> 1, E(w) will be
small because in eq(6.28) there are many oscillations of the factor exp(-iwt) within the
smooth variation of E ( t ) . Thus we see that in collisions with a simple harmonic oscillator of
frequency w, there is a natural cut-off to the energy transfer at a maximum impact parameter
----
Substituting eq(6.31) into eq(6.27), and restoring our notation of subscript 2 for the
target and subscript 0 for the incident velocity, we obtain the energy transfer in a straight-
Figure 6.6: The electric field components in a straight-line collision.
Notice that this is essentially the same expression as in eq(6.20) for the energy transfer t o a
free electron, except that the lower impact-parameter cut-off is not present here because of
the assumption of a straight-line orbit for the projectile, which is unjustified a t small impact
parameters. The rate of energy loss is then obtained, as before, by integration over impact
parameters from the minimum t o the maximum corresponding to the limits of applicability
of eq(6.31)
6.2.3
Classical Energy Loss Rate Formula
One final consideration is needed before we have an energy loss formula useful for practical
purposes. We have to have some way of applying the idealized harmonic oscillator calculation
to actual atoms. An atom in general has a number 2,say, of electrons bound to the nucleus.
Each electron may act as a target oscillator for energy transfer, and actually each electron
may act as one of an infinite set of oscillators, corresponding t o each of its possible quantum
transitions. Energy transitions of magnitude &, correspond to oscillators of frequency wi =
&,/7i, of course. To the i t h transition may be assigned an oscillator strength, f,, defined as the
ratio of the actual rate of energy absorption by that transition to that of a corresponding
harmonic oscillator. The semi-classical argument is then that each electron spends some
fraction of its time behaving as if it were each of the possible oscillators, and consequently
C f, = 2.There is a more rigorous theorem in quantum physics called the (Thomas-ReicheKuhn) f-sum rule which states that the sum of all possible transition oscillator strengths
from a specific level is equal to the number of electrons in the level. If this were applied
blindly t o all the electrons of the atom, it would give the same equation.
To obtain the total energy loss rate arising from collisions with a density of atoms n,,
whose atomic number is Z,we add up the contributions from all the possible transitions,
weighted by the oscillator strength of that transition. Thus we obtain for the logarithmic
term:
where we have defined a kind of average oscillator frequency ( w ) by the equation
The total classical energy loss rate is then
where we have substituted electron charge and mass, and for brevity denoted the argument
of the logarithm by
A=- YVo
.
(6.38)
(w)bgo
Actually, it turns out t o be possible to evaluate the Fourier transforms of the relativistic
fields in eq(6.30) in closed form and carry through the integration of the modified Bessel
functions thus obtained [ref6.2.3]. When that is done, two very small corrections to our
formula appear. The argument of the logarithm is multiplied by the factor 1.123 and an
additional relativistic term is added, equivalent to the replacement
Neither of these corrections is quantitatively significant. The result was first obtained by
Bohr in 1913, prior to the development of quantum mechanics. It is hardly complete as it
stands, since the average ( w ) has to be estimated. However, because ( w ) appears only in the
logarithm, even a rough estimate, for example setting & ( w ) equal to the atom's ionization
potential, will give a useful quantitative formula for the energy loss.
6.2.4
Quantum effects on close collisions
For the classical minimum impact parameter bgOto be applicable requires that the particles
of the collision behave as point particles down to that impact parameter. However, quant u m mechanics teaches us that particles do not behave like perfect points. The Heizenberg
uncertainty principle states that the particle is localized only within a position uncertainty
Az if its momentum uncertainty is A p such that A z A p 6.Alternatively one can say that
a particle with momentum p = y m v behaves like a wave with wave-vector k = p/&. Or
again, one can say that orbital angular momentum is quantized in indivisible units of 15. All
of these are ways of indicating that in collisions the effective position of a particle is spread
--
out over a distance of order filp. Consequently, quantum effects prevent us from extending
the classical integration over impact parameters below a value of
The classical bgOlower impact parameter cut-off will be applicable only if
where a is the fine structure constant, approximately 11137, This criterion is a requirement
that the collision velocity with electron targets should be less than Z1c/137.
In practice this means that electrons with energy greater than 1.9 keV, protons with
energy greater than 3.5 MeV, or alpha particles with energy greater than 55 MeV will not
be appropriately treated using the classical lower impact parameter cut off. Instead, an
approximation to the quantum-mechanical result may be obtained by simply cutting off the
impact parameter integration a t b, rather than bgO. If we choose3 b, = f i / 2 y m e v , then in
collisions of heavy particles with atoms, for which m, = m e ,
This value is then consistent with that obtained for the relativistic case using a quantum
scattering treatment and the first Born approximation, by Bethe (1930),
where again the final term, v;/c2, which we have not derived, is at most a small correction.
If the projectile is an electron or positron, then the quantum cut-off must be estimated
in the center-of-mass frame, and the expression becomes
6.2.5
Values of the Stopping Power
We have so far left open the question of what value to take for fi(w). Bloch (1933) showed
from an analysis of the Thomas-Fermi model of the electron charge distribution in an atom
that one would expect that fi(w) K Z . In recognition of the work of Bethe and Bloch, eq
6.43 is often referred to as the Bethe-Bloch formula. The formula is often written as
"
3The factor of 2 here is our only real artifice. It gives the argument of the logarithm equal to that obtained
by a full quantum calculation
with the quantity B, called the "atomic stopping number", corresponding to the factor
Also BIZ is then called the "stopping power" per (atomic) electron, recognizing that an atom
has Z electrons. The stopping power is determined from experiments, and the appropriate
value t o use for 5 ( w ) is determined from those measurements
A complication that we have not discussed arises because our treatment has assumed that
the orbital velocity of the electrons in the atom can be ignored relative to the velocity of the
incident particle. This is not the case when dealing with the inner shell electrons of high-Z
atoms or very low incident-energy projectiles. Then a reduction in the stopping number
occurs because (for example) the (innermost) K-shell electrons are ineffective in removing
the projectile's energy. This effect is numerically compensated by substracting a correction
In this form, the value of 5 ( w ) is empirically determined to be about 11.5 x Z eV, and CK
is a function of the quantity E = (c2/v,2)(Z 0.3)2a2 (which represents the squared ratio of
the K-shell velocity to the projectile velocity). A simple approximate form for CK is
-
-
correct t o within 10% from E = 0 to E = 2. It tends to zero at high projectile energy and
1. These and many other details have been
peaks at about unity at low velocity, where E
reviewed by Evans (1955).
6.26
Effects of surrounding particles on distant collisions
Let us return now to our primitive energy loss rate calculation, eq 6.34 which may be
considered in the form
47r
(6.47)
d!
In the preceding sections we have been discussing appropriate choices of b,
based either
on the classical effects of large scattering angles (giving bgO) or on quantum-mechanical
effects of the de Broglie wave-length of the projectileltarget combination. We also discussed
based on the effects of the binding of electron targets to their nuclei.
the appropriate b,
However, another effect can sometimes be more important than the atomic binding structure
in determining bm, namely the influence of surrounding particles.
We have tacitly assumed so far that the interaction of the projectile and any specific target
can be treated zgnorzng the effects of the other targets in the vicinity. We have calculated the
projectileltarget interaction in isolation and then presumed that we can add up the effects of
all the different targets via a simple impact-parameter integration. This may not be the case.
For example, it definitely is not the case when the electrons of the target are unbound; or in
other words for a plasma target. In that case there is no intrinsic cut-off t o the the collision
integral arising from the oscillator effects introduced in section 6.2.1 and the effect of the
nearby particles essentially always determines bm. Even in collisions with atomic matter,
especially for relativistic electrons, the effect of nearby particles can significantly lower the
energy transfer rate. In the atomic collision context the corrections are often referred to as
the "density effect" because they are most significant for high-density matter.
It is still the case that transfer of energy to the target arises from the electric field
produced by the incident projectile. However, what we need to do is to account for the
influence of the other particles in the target medium on the electric field that the projectile
produces at a specific target. Expressed in this way, it is immediately clear that what we
need is to take account of the dzelectrzc propertzes of the target medium. The individual
particles of the medium respond t o the influence of charge (the projectile in this case) so as
to alter the electric field in the medium from what it would otherwise have been. This is
exactly what we mean by the dielectric response of the medium.
Of course, though, it is not the steady-state dielectric response that we require but the
response at the high frequencies of interest in the collisions. Moreover, when we think
about a target medium consisting of a density of idealized oscillators, as we did before, it
is the properties of those oscillators themselves that determines the dielectric response at
frequencies close to their resonant frequencies. Thus the dielectric response and the energyloss collisional response are not two separate properties of the medium, they are intimately
connected.
The idealized oscillator model can be generalized to discuss a medium with any relative
dielectric permittivity t(w) having a resonant form ( t 1 K (w w,)-l), and an expression
for the rate of loss of energy of an incident projectile to this resonance can then be obtained.
Fermi (1940) first gave the following formula, which would take us too long to rederive, for
the energy loss attributable to collisions with impact parameter greater than a as
-
where R denotes real part,
argument is s such that
0 = vole, K1 and K2 are
-
modified Bessel functions, and their
a2w2
s2 = -[I
P2t(w)]
(6.49)
v,"
It can be shown, but not trivially, [Jackson] that this dKld! reduces to the the Bohr expression (eq 6.39) if the P2t(w) term in s is neglected.
Rather than pursue the topic for the atomic case, let us consider a simple argument for
a plasma. The dielectric constant for a (magnetic field-free) plasma at high frequency is
-
where
is called the plasma frequency. Therefore when the field frequency of interest is less than wp
the dielectric constant is negative and wave electric fields no longer propagate in the medium;
instead they decay exponentially with distance from their source. In collisions, as we have
seen before, the frequency of the interaction electric field is approximately vo/b. Therefore,
for impact parameter, b, greater than vo/wp we would expect that the effectiveness of the
we
collisions would fall off because of the dielectric effects. Applying this value for b,
obtain an energy loss rate expression corresponding to eq 6.37 as
but with A given approximately by
What we have done, in effect then, is t o replace the value ,b,
of A, eq (6.38) with
vo
bm, = .
= yvo/(w) in
the definition
(6.53)
The factor by which the logarithmic argument A of the Bethe-Bloch formula is multiplied
is therefore ywp/(w). But the density effect can only lower the absorption rate so we should
= min(vo/wp, yvo/(w)). The electrons behave as if they are
more properly have used ,b,
free when w > wij (w). Hence plasma-like, i.e. free-electron, behaviour occurs only when
wp > (w), which is when the plasma expression for ,b,
applies, because it is the smaller.
A rough estimate of the ratio of wp/(w) may be obtained by taking the density of atoms
in a solid t o be about lo3' m-3, and the electron density to be 2 times that. Then
-
-
112 eV so we expect the plasma effect to be
For medium weight solid elements, fi(w)
slightly noticeable since on this basis wp/(w) > 1. The question is a little more complicated
than this, though because not all the electrons are going to behave as if free so we have
somewhat over estimated the density of the electrons that behave as if free. In extreme
relativistic cases, y >> 1 the plasma (density) effect will always dominate.
6.3
Angular Scattering from Nuclei
Up t o this point we have been discussing the energy loss of the projectile and have focussed
on its interactions with electrons. This focus on electron targets is entirely appropriate for
calculating energy loss because, as illustrated by eq (6.21) or (6.34) the rate of energy loss
is, classically, inversely proportional to the mass of the target particle4. Therefore the loss
of energy is in fact predominantly to the light particles, electrons, and this predominance
4This proportionality can be traced to the inverse dependence of the energy transfer in a collision on m s ,
but only because cancellation of reduced mass factors occurs in the product Qb&.
depends only on the elementary dynamics of collisions. However, in addition to losing energy,
the projectile also generally experiences angular scattering in the direction of its velocity. If
this angular scattering is our concern, as it was in Rutherford's original experiments on the
angular scattering of alpha particles which established that the nucleus is far smaller than
the atom, then collisions with the heavy particles in our scattering medium, the nuclei of
the atoms or the ions of a plasma, are important. This process, illustrated in Figure 6.7,
is often called "elastic scattering", although the expression may be considered somewhat
misleading in that some energy is lost by the projectile in the collision, and the process is
no more elastic than a collision with a free electron, for instance.
[email protected]
\Electron Cloud
Figure 6.7: Angular scattering from nuclei occurs only if the impact parameter is less than
the size of the electron cloud.
Qualitatively, the relative importance of energy loss and angular scattering can be grasped
by imagining the difference between a ping-pong ball colliding with a random arrangement of
snooker balls, or a snooker ball colliding with a random arrangement of ping-pong balls. In
the first case, the light projectile will bounce around changing its direction of motion many
times before losing its energy; while in the second case, the heavy projectile will plough
through the light targets, losing energy faster than its direction is deflected.
The angular scattering of a particle in a classical coulomb collision is governed by the
Rutherford formula for the differential scattering cross-section per unit solid angle at a
scattering angle in the center of muss frame, x,,
This formula may readily be derived from the considerations in section 6.1.1. It shows that
the predominant scattering is through small angles. Those small angles arise from large
impact parameters. There are some collisions, of course, which arise from small impact
parameters, close to bgO,that give rise to large scattering angles, but these are far fewer in
number than the small-angle collisions; so by the time the probability of scattering by a large
angle is significant, multiple scatterings by small angles will have caused a kind of diffusion
of the direction of the particles in perpendicular velocity. Figure 6.8 illustrates an idealized
situation, in which the projectile loss of energy is taken as zero, so its velocity vector has
constant magnitude and moves on a sphere. Taking the initial direction to be along the
z-axis, each small-angle collision causes a random step t o be taken in the (v,,v,) plane.
vyl
initial v
Side View
End View
Figure 6.8: Multiple small-angle Coulomb collisions cause a diffusive "random walk" of the
angle of the projectile velocity, or equivalently its perpendicular components.
Setting the large-angle collisions aside for a moment, we can treat the total angular
scattering experienced by a projectile passing through a finite length of scattering path as
the result of many small scatterings each of which has random direction and magnitude,
governed by the fact that cot(x,/2) = b/bgo (eq 6.17). Although we cannot calculate for any
individual projectile what its final angle will be, we can treat the whole process statistically,
by presuming there to be many small-angle scatterings. Actually, this calculation requires
not the Rutherford differential cross-section per unit solid-angle a,but the differential crosssection per unit scattering angle x,,
which result is obtained immediately from our previous formulae.
The mean scattering angle experienced by the projectiles is always zero, because there
is equal probability of scattering at negative and positive angles; the scattering is isotropic
in the (v,,v,) plane. The spread of scattering angles is quantified by the mean square
scattering angle, which is non-zero and can be evaluated as follows. Succeeding collisions are
statistically independent of each other. The final value of v, is given by the sum of the steps
in v, a t each of the individual collisions. (Similarly for v,.) We therefore make use of the
basic statistical theorem that the variance (which is the mean-square value for a zero-mean
random variable) of the sum of independent random variables is the sum of the variances.
We perform this sum by dividing the collisions into appropriate ranges of scattering angle
dxC and azimuthal angle d4. The number of steps per unit path length belonging in these
ranges is
dff
d4
d N = n2-dxC(6.57)
dxC 2.rr
and the change in v, that such collisions cause is
6v,
=
(m,/ml)vo sin X, cos 4
.
(6.58)
Here, the quantity (m,/ml)vo is the initial (and final) speed of the incident particle (1) in
the center-of-mass frame. Consequently, the total variance of v, per unit path length arising
from all possible types of collisions is
Performing the integration over azimuthal angle, ,
cross-section from eq (6.56) we get
and substituting for the differential
The final integral may be transformed using trignometric identities, becoming
/
sin2 xccsc2(x./2) cot(xr/2)dxC= 8
/
1
- -
S
sds
(s = sinxc/2).
(6.61)
The upper limit of the integral is s = 1. The singularity of this expression at zero lower
limit of s shows again the now-familiar need for a cut-off of the collision integral at large
impact-parameter (small X, or s). That cut-off and eq(6.19) make the value of the integral
, )1, where ,b,
is the maximum impact parameter, and the term should be
8(ln b,,/bgo
dropped since it is an artifact of the approximation implied by our use of eq(6.58). In the case
of scattering by a plasma, the relevant impact-parameter cut-off is the length beyond which
the collective interactions in the plasma screen out the electric field of individual nuclei. This
distance is called the Debye length. When the scattering is from neutral atoms, the relevant
cut-off length corresponds t o the size of the atom, because for impact parameters larger than
the atom the projectile sees the whole atom, neutral because of its electrons, rather than a
bare nucleus.
An identical treatment governs the y-component v,, and consequently the square of the
total transverse velocity v l = vz vz evolves as
+
approximately the size of the atom. For small angles 6' zz vl/v and so this equation
with ,b,
can be written in terms of the angle of the scattered velocity direction:
After a finite path length
!,
there is a distribution of v l with variance
which we assume is still small compared to vi so that small-angle approximations remain
valid. Because this distribution arises from many independent scatterings, it becomes Gaussian (following the Central Limit theorem of statistics):
with (vz)given by eq(6.64). We may alternatively regard the Gaussian shape as arising because the particle distribution is experiencing a dzffuszon of velocity from an initial localized
distribution (delta function) at v l = 0. The solution of the diffusion equation in this case
is this Gaussian.
The maximum impact parameter (minimum x,) is determined by the shielding of the
nucleus by its atomic electrons. Only for impact parameters small compared to the atom
size, will the projectile see the bare nucleus because then it penetrates deep inside the electron
is approximately the radius of the electron cloud surrounding the
shielding cloud. So b,,
nucleus. This is generally taken to have a characteristic size approximately5 a o / ~ 1 / 3 .
There is no mathematical compulsion to cut off the upper limit of the X, integral short
of xC= T, that is s = 1. However, if very energetic particles are involved, the value of bgO,
which is inversely proportional t o particle energy, becomes very small, eventually so small
that it is smaller than the size of the nucleus. In that case, the large-angle scattering is
affected by the structure of the nucleus itself and the upper limit is affected. Of course,
this is the basis for experimental high-energy physics investigations of nuclear structure by
electron scattering, but it requires electron energies greater than roughly Z e 2 / ( 4 ~ t o r , )(-- Z
MeV), where r, is the nuclear radius, of order 10-15 m, and Z its nuclear charge.
6.4
Summary
Collisions of charged particles are governed by the long range Coulomb force. The range of
that force is limited by one of several different processes, depending on the exact physical
A minimum impact parameter for the prosituation to a maximum impact parameter b,,.
cess is also needed if approximations such as that the collision has a straight-line trajectory
are made, or if quantum effects are important. Table 6.1 gives a summary of the situations
discussed.
'See M.Born "Atomic Physics" 8th ed., Blackie pl99, for a derivation of the Thomas-Fermi distribution
of electron density around an atom based on the Pauli exclusion principle and a continuum approximation.
1
1
Impact parameters
Collision Type
bmn
Classical Coulomb
bgo
Stopping Power (per e1ectron)I
In A = BIZ
b,
7
~
0
1
ln Y"O
wbso
~
Quantum ion loss t o atoms
Corrected for inner shell effects
002
1 123700
Classical energy loss to atoms
" filymv
yvolw
ln 1-
Quantum electron loss to atoms
- filymv
Density effect (non-rel. plasma)
bgo
Angular scattering from nucleus
bgo
yvo/w
~O/WP
1
lnl(w)hsol-~ ln h ( ~ ) 1 7
0;
2~2mev;
(
-
)l
2
cklz
1 5w027 *
6 , ~ )
-ao/~1/3 Table 6.1: Summary of collision calculations
In collisions of the projectile particle 1, initial velocity vo, with particles of type 2, density
n 2 ,the rate of loss of kinetic energy K per unit pathlength !is given by
and the angular scattering from nuclei by
with the 1nA values indicated. See eqs(6.18) and (6.3) for other definitions
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